Is there a typical structure for a fantasy trilogy?
To be more specific, would they usually follow the three-act structure, with an act per book? And would each book need a self-contained story arc with a proper beginning and ending? Or could you get away without a proper resolution until the end of the third book?
I know this could be subjective, but what are the norms when approaching this?
There are two main ways to structure a series: each book is essentially a stand-alone with a continuing story as part of the plot (Harry Potter), or each book is a critical part of the whole and they are difficult to read out of sequence or without the other books (Lord of the Rings). Either is fine; they just accomplish different things.
Stand-alone books have the benefit of being easy to draw in new readers out of sequence, as Shan observes. They don't have an overall story to tell as much as they are a journey being taken with the same characters. Harry's story does have a "proper resolution" in the long view, but each book resolves its individual crises, particularly the first three.
A series of books is one story being told in many parts. Each book does need some kind of "beginning, middle, and end," but by no means does each book require closure. Fellowship of the Ring ends with the Fellowship broken: one of the Nine Walkers dead and one presumed dead, two kidnapped, and two vanished off to Mordor. But the story has progressed from "One Ring's discovery" to "Frodo accepts the quest" to "Frodo and Sam leave for Mordor." The "proper resolution" doesn't occur until the multiple ends ;) of Return of the King.
I agree with Shan that if I'm in a bookstore and I see "Book 1 of 3" I may not pick it up unless Books 2 and 3 are next to it on the shelf. However, don't let that stop you from telling your story in the way that your story needs to be told. Write the best story you can, structure it however makes it the most appealing, and let the marketing folks at the publisher worry about selling it in pieces.
David Eddings originally planned his Belgariad series as a trilogy, and the publisher broke it into five books. George R.R. Martin planned the Song of Ice and Fire story for four novels, which grew into six and then split into seven. Yep, pain in the butt for us readers now, but when it's all over, they will all be together on one shelf for future generations.
Every part of your work needs to have its own logical arc and structure. In a novel, this can be a chapter or a scene and in a trilogy or series this a book. Each piece should have a beginning, middle, and end. The parts don't live in isolation, though, and each one should leave give the reader enough to enjoy what they are reading but hold back enough to leave them wanting more.
Think about the Empire Strikes Back. By the end of the movie, all the immediate threads have been wrapped up (everyone is rescued, the Empire's motivation is made more clear, and Luke's got a new daddy). The nerds were satiated to the point where they didn't start ripping chairs out of the floor and start tearing down screens. But practically all the big picture stuff was left unresolved and the nerds were going to have to wait three long years to get the full resolution.
Update:
Since it has been mentioned in several responses I think there is one more thing to address: Just because the sequels had not been written at the time Star Wars produced doesn't mean that Lucas didn't already have the ideas marinating in his bearded head. And that's an important point for two reasons:
I personally don't like books that force me to buy another 2 books to get the whole story (and maybe wait months or years if the books haven't been published yet).
More and more, whenever I see something like "Part 1 of the Foo Boy trilogy" I just put the book back on the shelf, unless the author is someone I know and enjoy really well(which is very rare).
I think expecting your readers to buy more books to get the whole story is a bit condescending. I prefer something like the "Hitchhiker's guide to the galaxy" series, where each book is independent, can be read separately.
So my advice is to carry elements of your first book into the second, like some of the characters, the Universe etc, but keep each book independent, so that readers can read any part and still understand the story.
You can have a common underlying theme or plot that runs through all the books, but each book must contain enough information to be read by itself. Like many people, I read the Harry Potter series out of order (at least the first few ones), but had no trouble understanding the whole plot. That, I think, is one of the reasons the series was so successful- I went back and read the other books because I enjoyed them, and not because the author left the story incomplete.
A very typical structure, not already mentioned, is to write a popular standalone book and realise afterwards that instead of starting afresh for your second book, you can write two more books in the same setting or with the same characters.
Perhaps this is more common in the world of film. Even Star Wars, an example given in another answer, wasn't conceived as part of a series until after it (wikipedia). It didn't become Episode IV: A New Hope until after The Empire Strikes Back was released.
I don't think there's anything wrong with this approach. It's hard to write even one good book. If you do happen to write one, and it turns out some of your characters are still alive at the end, then it's time to consider whether you can add another book or two, and what structure the series will have. If you start out trying to write a trilogy, you may find you've bitten off a lot more than you can chew.
